Implicit differential equations and vector fields with non-isolated singular points

Vector fields with singularities that are not isolated, but form a smooth submanifold of the phase space of codimension 2 are studied. Fields of this kind occur, for instance, in the analysis of implicit differential equations. Furthermore, under slight perturbations of the original problem the variety of singular points does not disappear or degenerate, but merely deforms. Results on the structure of invariant manifolds of such fields are obtained, along with smooth normal forms for certain cases